Chapter 1, Problem 37CRP

a.

Explanation of Solution

Express $-\frac{7}{2}$ using the 8-bit floating point format.

The process to encode the values using the 8-bit floating point format is,

• Convert the given value in the binary representation.
• Copy the bit pattern into the mantissa field from left to right start with 1.
• Copy the bit pattern into the mantissa field from left to right start with 1.
• The exponential field is determined by imagine the pattern of mantissa field with a radix at its left and obtain the original binary number by determining the number of bits and direction of radix that must be moved.
• The remaining three bits represent the exponential field that is shown in the below table.

Bit pattern	value
$111$	3
$110$	2
$101$	1
$100$	0
$011$	$-1$
$101$	$-2$
$001$	$-3$
$000$	$-4$

• When the given value is positive then the bit assigned on high order bit is 0 and when the given value is negative then the bit assigned on higher bit is 1.
• Convert the number $-7$  into binary equivalent.

$\begin{array}{ccc}2& 7& \\ 2& 3& 1\\ 2& 1& 1\\ & 0& 1\end{array}$

• The binary representation is the sequence of the remainder from down to bottom

b.

Explanation of Solution

Express $\frac{1}{2}$ using the 8-bit floating point format:

The process to encode the values using the 8-bit floating point format is,

• Convert the given value in the binary representation.
• Copy the bit pattern into the mantissa field from left to right start with 1.
• Copy the bit pattern into the mantissa field from left to right start with 1.
• The exponential field is determined by imagine the pattern of mantissa field with a radix at its left and obtain the original binary number by determining the number of bits and direction of radix that must be moved.
• The remaining three bits represent the exponential field that is shown in the below table.

Bit pattern	value
$111$	3
$110$	2
$101$	1
$100$	0
$011$	$-1$
$101$	$-2$
$001$	$-3$
$000$	$-4$

• When the given value is positive then the bit assigned on high order bit is 0 and when the given value is negative then the bit assigned on higher bit is 1.
• Convert the number $\frac{1}{2}$  into binary representation

c.

Explanation of Solution

Express $-3\frac{3}{4}$ using the 8-bit floating point format:

The process to encode the values using the 8-bit floating point format is,

• Convert the given value in the binary representation.
• Copy the bit pattern into the mantissa field from left to right start with 1.
• Copy the bit pattern into the mantissa field from left to right start with 1.
• The exponential field is determined by imagine the pattern of mantissa field with a radix at its left and obtain the original binary number by determining the number of bits and direction of radix that must be moved.
• The remaining three bits represent the exponential field that is shown in the below table.

Bit pattern	value
$111$	3
$110$	2
$101$	1
$100$	0
$011$	$-1$
$101$	$-2$
$001$	$-3$
$000$	$-4$

• When the given value is positive then the bit assigned on high order bit is 0 and when the given value is negative then the bit assigned on higher bit is 1.
• Convert the number $3$  into binary equivalent.

$\begin{array}{ccc}2& 3& \\ 2& 1& 1\\ & 0& 1\end{array}$

• The binary representation is the sequence of the remainder from down to bottom.

${\left(3\right)}_{10}={\left(11\right)}_2$

• Convert the number $\frac{3}{4}$  into binary representation

d.

Explanation of Solution

Express $\frac{7}{32}$ using the 8-bit floating point format:

The process to encode the values using the 8-bit floating point format is,

• Convert the given value in the binary representation.
• Copy the bit pattern into the mantissa field from left to right start with 1.
• Copy the bit pattern into the mantissa field from left to right start with 1.
• The exponential field is determined by imagine the pattern of mantissa field with a radix at its left and obtain the original binary number by determining the number of bits and direction of radix that must be moved.
• The remaining three bits represent the exponential field that is shown in the below table.

Bit pattern	value
$111$	3
$110$	2
$101$	1
$100$	0
$011$	$-1$
$101$	$-2$
$001$	$-3$
$000$	$-4$

• When the given value is positive then the bit assigned on high order bit is 0 and when the given value is negative then the bit assigned on higher bit is 1.
• Convert the number $\frac{7}{32}$  into binary representation

e.

Explanation of Solution

Express $\frac{31}{32}$ using the 8-bit floating point format:

The process to encode the values using the 8-bit floating point format is,

• Convert the given value in the binary representation.
• Copy the bit pattern into the mantissa field from left to right start with 1.
• Copy the bit pattern into the mantissa field from left to right start with 1.
• The exponential field is determined by imagine the pattern of mantissa field with a radix at its left and obtain the original binary number by determining the number of bits and direction of radix that must be moved.
• The remaining three bits represent the exponential field that is shown in the below table.

Bit pattern	value
$111$	3
$110$	2
$101$	1
$100$	0
$011$	$-1$
$101$	$-2$
$001$	$-3$
$000$	$-4$

• When the given value is positive then the bit assigned on high order bit is 0 and when the given value is negative then the bit assigned on higher bit is 1.
• Convert the number $\frac{31}{32}$ into binary representation.